Research Laboratory of the Portland Cement Association Bulletin 6 Dynamic Testing of Pavements BY GERALD PICKETT April, 1945 CHICAGO Authorized Reprint from Copyrighted Journal of the American Concrete Institute New Center Building, 7400 Second Boulevard Detroit 2, Michigan April 1945* Proceedings F. 41, p. 473 * TitU 41-20 B part of PROCEEDINGS, A O 1 C IRETE I TE Vol 41 AMERICAN JOURNAL < the CONCRETE INSTITUTE . Vol 16 No 5 M » jAN April 1945 Dynamic Testing of Pavementt Bv ALD PI ■ SYNOPSIS n 1 •1 e u.. t i ■ i - : J • I K' "i INTRODUCTION I , -1 • I I ' arm t a ■ i « I i 7 \ if i 1 1 ! I ' 474 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE April 1945 40 cycles per second. Recently, Long, Kurtz, and Sandenaw^ 67) have developed instruments that give reasonable accuracy for the time of travel of waves between two points on the surface of a pavement slab, even when the points are no more than a foot apart. They measured velocity of waves produced by impact and also those produced by sustained vibration at frequencies ranging from 1000 to 2000 cycles per second. They also made some progres- in interpreting the results in terms of Young's modulus of elasticity of the pavements by the use of equations developed by Lamb (S) and Timoshenko. (9) The accomplishments noted above and the fact that the use of the equations of Lamb and Timoshenko for this purpose may be question- able stimulated the author to prepare the present paper. LIMITATIONS OF PREVIOUS EQUATIONS The equations of Lamb and Timoshenko are not strictly applicable to pavements because: (1) They are for freely vibrating slabs, whereas the pavement is in contact with the subgrade. (2) They are for a two-dimensional problem (plane waves), wherea- the vibration of a pavement is a three-dimensional problem. In general both the shape and the support of field structures are such as to make analysis of vibration more complicated than that for small laboratory specimens. So far as is known, the nearest analytical approach to the vibration of a pavement was a study made by Love a0) in which he investigated the vibration of the earth's crust, a — uming that the elastic properties of the crust differ from those of the interior. However, Love's studies were also confined to two-dimensional problems and only dealt with cases in which the crust was less rigid than the interior, whereas a con- crete pavement will usually be more rigid than its subgrade. SCOPE OF PRESENT DISCUSSION This paper will discuss certain possible modes of vibration of a pave- ment in contact with it- s ubgrade that are likely to occur in the dynamic testing of pavements. An equation giving the relation between driving frequency, thickness of slab, elastic constants of both pavement and -ubgrade, and the velocit\ T of wave is derived on the i -sumption of con- tinuity of motion between pavement and subgrade. The equations of elasticity applicable to a homogeneous, i-otropic, elastic solid are used for both the slab and the subgrade. If experiments prove that the equations are generally applicable to pavement- in place, then from a few dynamic measurements it should DYNAMIC TESTING OF PAVEMENTS 475 be possible to determine not only Young's modulus for the concrete but also the thickness of the pavement and a modulus for the subgrade. PARTICULAR SOLUTIONS OF THE DIFFERENTIAL EQUATIONS OF VIBRATION When expressed in the cylindrical coordinates r, 0, z the differential equations of vibration for a homogeneous, isotropic, elastic solid are: (n) d r r dQ dz dt r dQ dz dr dt 2 nrVt dA 2G d , m , 2G dU d 2 w X + 2G)— -(r\) + — — = p — dz r dr r dQ dt- where u, v, w are displacements in the r, 9, z directions, respectively; t is time; du . u . 1 dv dw A — H H — ; -r 21 dr r r dQ dz 1 dw dv r dQ dz du dw 2T = — ; dz dr dv . v 1 du 2W = — + dr r r dQ p = mass density; X = — , Lamp's constant; (1 + M ) (1 - 2 M ) Q = . , modulus of elasticity in shear; 2(1 + m) E — Young's modulus; and ^ = Poisson's ratio. Four particular solutions of the differential equations will be used in the following discussions. Two of these, designated H and V, apply to the pavement slab, and two, designated Hi and Vi apply to the subgrade. The solutions will be used in pairs, the H solution for the pavement with the Hi solution for the subgrade, etc. In the H and Hi solutions points on the line r = move in a horizontal direction and in the V and Vi solutions these points move in a vertical direction. In the following all subscripts of unity except that designating a Bessel 476 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE April 1945 function of the first order refer to the subgrade The solutions are as follows: (J and Ji are Bessel functions.) u Z sin 6 cos pt v Z cos G cos pt Jo (a r) Jx (ar) ar w = Z' sin 9 cos pt J\ (ar) J\ (ar) ar (H) U>! t'l IV I Z x sin cos pt = Zi cos 9 cos pt Jo (ar) Ji (ar) ar = —Z x ' sin cos pt Jx (ar) Ji (ar) ar _ (HO u v w = Z cos pt Jx (ar) = = — Z' COS pt J (ar) (V) u x Vi Id = Z\ cos pt J\ (ar) = = Zx COS pt J (ar) (Vi) where Z = M cosh mz + 3/' sinh mz + N cosh nz + TV' sinh nz\ Z' = Mm . . . M'm , Na . , sinh mz + cosh mz + — sinh nz a a n V + - — cosh nz; n Zj = Re{Mxer m { + AW] ; Z/ = #e -Uimi „ . . Nia m _ a ni hi n = aVl a > a = aVl P b 2 mi n a > X + 2G ' Qi b = V p G 6i aVl - a i 2 ; aVl ~ bi 2 ; P J Pi P I Pi + 2G l * \ DYNAMIC TESTING OF PAVEMENTS 477 p = 2tt times the frequency of vibration; a = p/V = 2w/l where T r is the radial velocity which the wave approaches asymptotically as it gets farther from its source and I is the corresponding wave length; and M, N, M f , N', M u and A\ are constants proportional to the amplitude of vibration. Re [ ] signifies the real part of the expression in the bracket if either mi or n x is imaginary in a mathematical sense. If both mi and m are real, then the Re in front of the brackets may be disregarded. In each pair of solutions, for example, H and Hi, the frequency p/2r, the velocity p/a, the six amplitude constants M, N, M' } N', M h and Nij and the physical properties are arbitrary. That is, the differential equations are satisfied for any arbitrary values of these parameters. However, the boundary requirements at the top of the pavement and at the common boundary between pavement and subgrade permit the elimination of the six amplitude constants. The result is an equation, called the frequency equation, giving the relation between frequency, velocity of wave propagation and the physical properties of pavement and subgrade. It is of interest that the frequency equation is the same whichever pair of solutions is used and depends only on the assumptions made in regard to boundary conditions. In the derivation which follows the assumption is made that the top of the pavement is free of force and that the boundary stresses and displacements of the pavement are equal to those of the subgrade at their common boundary. DERIVATION OF THE FREQUENCY EQUATION The plane z = is taken in the middle of a slab of thickness 2c and the direction G = is taken as clue east as shown in Fig. 1. The assumed boundary requirements result in the following relations:* 1. The top of the pavement is free of vertical force. a z = at z = —C. 2. The top of the pavement is free of radial and tangential forces. r« = r 02 = at z = - c. 3. At the common boundary the vertical displacement of the pave- ment equals the vertical displacement of the subgrade. w = Wi at z = c. i. At the common boundary the radial and tangential displacements of the pavement equal the radial and tangential displacements, respec- tively, of the subgrade. u = u x and v = r, at z = c. 5. At the common boundary the vertical normal stresses are equal. <r, = a\, at ,: = c. *Two relations are given in each of requirements 2, 4. and 6; but in each case the second relation is satisfied if the first is satisfied. Therefore the two relations give only one independent equation. 478 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE April 1945 Fig. 1 — Element of pave ment slab and subgrade North West South Subgrade !/ "T 6. At the common boundary the boundary shear stresses are equal. ?Tt = Tirz and t Qz = 7i e * at z = c. When either pair of solutions for displacements is used and use is made of the relations previously given between m, n, X, G, a, b, m u ni, Xi, G h ai, and 5i and of the following relations between stresses and displacements, °* = X ( dw du u dz dr r r 1 dv\ r dO/ + 2G dw 3z TZ Qi dw du dr t& = G dz dw dv r <99 dz the six boundary relations listed above result in the following six equa- tions, respectively: (2 — ¥)[M cosh mc — M' sinh mc] + 2 [N cosh nc — N' sinh nc] = . . (1) 2m [ — M sinh mc -J- M' cosh mc) a. + (2 - b 2 ) — [ - N sinh nc + N' cosh nc] = n JbfWt — [M sinh mc + M' cosh mc] + — [TV sinh nc + TV' cosh nc] a n 7?c mi . , TV — er m l e H ni k J (2) (3) DYNAMIC TESTING OF PAVEMENTS 479 M cosh mc + M' sinh mc + .V cosh nc + W sinh nc = fie [ Mie-Y + iV>V ] (2 - b 2 ) [M cosh wc + 3/' sinh mc] + 2[iV cosh nc + iV' sinh nc] (4) G?! G fie [ (2 - 6r) Mie-V + 2AV-Y ] (5) 2m a \M sinh mc + M' cosh mc] + (2 - 6 2 ) — |JV sinh nc + i\T cosh ncl (?i G fir a a (6) 2 ^- l e-*V + (2 - fei 2 ) - tfi *•« The elimination of the six amplitude constants M, AT, iV, iV', M u and JVi from these six equations gives the following frequency equation: 1 - A, - A* Wi (1 .4, - A/) 3 1 - A i,' 1 - A A (^; ' where A x = 4> 2 — b 2 2 coth mc — k coth »c + - 2 6r & 2 V 1 a <> 2 2 / 1 A -r 2 ~ &i a 2 - b 2 cQtb _ 2 - 6x* k CQth M 2 2 2 6 2 + fo —VI - <$ A 3 = <}> 2 -_W 2^ coth mc _ AzilL fc coth nc _ 2 b 2 2 2 + j\ ^ VI — 6 2 coth mc coth nc 2 A 4 = '9 I>2 _ coth mc - k coth nc L 2 + 2 - 6r b 2 Vl - V 2 2 fei coth mc coth nc 4> = G\ 2G 2 - 6 2 coth mc — A: cotli nc 2 h = V(i - a 2 ) (i - & 2 ) ; ft 1= = VI - &i 2 ^ V" < x > i- e -' if til is real / h = - V6i- - 1 cot (acV&i 2 - 1) i f fri 2 > !> i- e -> if "i is imaginary /i= VI a if ai 2 < 1, i.e., if mi is real; f 1 = V«r - 1 tan (acVoti 2 - 1) if <*i 2 > 1> ie., if mi is imaginary 480 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE April 1945 The expressions for Ai, A 2 ', A 3 ' } and A,' used in the frequency equa- tion are the same as for the expressions for A u A 2 , A&, and A h respec- tively, except that in every instance coth is replaced by tanh. THE MEANING OF IMAGINARY AND COMPLEX VALUES FOR m, n, mi, AND m The question of real and imaginary quantities entered into the solu- tions for the subgrade. The reason for this was that certain of the expressions became complex (contained both real and imaginary parts) under some conditions. If the velocity of propagation of the waves horizontally is greater than the normal travel of disturbances within the subgrade, then either rii or both n* and m% become either imaginary or complex quantities in a mathematical sense. If internal friction is neglected as in the present study, then they become imaginary, but if internal friction is taken into account they become complex quantities. That is, if in the foregoing development p/a is greater than V <?i/pij then 61 will be greater than unity and rti will be imaginary, and if p/a is also greater than V (Xi + 2Gi)/pi, then mi will also be imaginary. When n x and m y are imaginary, the expressions €r\ z and e~ m i become trigonometric functions of the deptli with both real and imaginary parts. For example, the real part of the expression for Z\ becomes Z x = Mi cos (az V aj - 1) + A r i cos (az V &i 2 - 1) and for Z/ becom< Z,' = Mi V ar - 1 sin (az V a x 2 - 1) - N> sin (az V br — 1) V 6r - 1 The factor n may also be imaginary, but such a possibility does not introduce any ambiguity into the solutions since cosh nz and (sinh nz)/n arc both real whether n is real or imaginary. The factor m will ordinar- ily not be imaginary because the .^liear modulus of the pavement is greater than the shear modulus of the subgrade. If either ~tn y or is imaginary, then as just indicated, a part of the expression for a displacement within the subgrade will be a trigonometric function of the depth z. Consequently, this part of the expression will indicate no diminution of maximum amplitude of vibration with increase of depth in the subgrade. Had internal friction been taken into account, the solutions would indicate a decrease in amplitude with increase of deptli. For example, if the assumption is made that internal friction is adequately considered by replacing the usual Hooke's law by equa- tions of the type* The term with /has been added to the usual expression for Hooke's law such as given by Timoshenko in Theory of Elasticity (New York: McGraw-Hill. 1934), pp. 7-10. DYNAMIC TESTING OF PAVEMENTS 481 ., +f de - l dt E a x — n<Jy — /da I xy I J dt G then solutions corresponding to those given in V, above for the subgrade can be written e ipt { tf (l) (ar) u l = Zi sin 6 Re l_ . a/ with similar explosions for n and w h where the "real" Bessel functions of the first kind (J and JO nave been replaced by "complex" Bessel functions of the third kind (#,<» and flV 1 *)*; the "real" function cos pi has been replaced by the "complex" function e ipt ; and m x and m have the values «i 2 i b{2 in place of the values given previously. A separation of im and m into their real and imaginary parts shows that each has a positive real part for all values of a L and b x . Therefore, when internal friction of the subgrade is taken into account, the ampli- tude of vibration decreases with increase, of depth. In the above type of solution for vibration with internal friction the displacements and stresses are discontinuous at the line r - 0. It is on this line that the energy necessary to maintain sustained vibration is assumed to be supplied. To restrict the driving force to points on the pavement rather than at the line r = would necessitate a still more com pli c at ed solution . Since it is believed that internal friction has only a small effect on the velocity of propagation of the waves and since it was desired to keep the analysis relatively simple, friction was neglected in the deriva- tion of the frequency equation. NUMERICAL SOLUTION OF THE FREQUENCY EQUATION The frequency equation derived above expresses the relation between the velocity! of wave propagation p/a, the frequency of vibration p/2r, and the physical properties of the pavement and subgrade. Un- fortunately, this relation (Equation A) is rather involved, and numerical solution is not made readily. The method of solution found to be best in general was as follows: *Bessel functions of first, second, and third ktabn treated I in Tables of Functions, by Jahnke and •tfmAc **H FH (New York- G. E. Stechert & Co., 1938), pp. izb-zbK. , W mifl L remembered that the velocity of waves traveling radially depends on distance from the urce. p/a is the velocity that they approach with increase of distance. source 482 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE April 1945 First, assume values for G%/G, X/G, X1/G1, pi/p. oond, select a value of b, the ratio of velocity of propagation to Third, determine a, a 1( and &i and the function s of these quantitii and <>f b that will b ded later such a- V 1 - 6 2 , V 1 - &i*, (2 - ft )/2, Pourtl lect a value of *c *, th< patio of frequency times pave ment tin. to the i ocity of wav< propagation. 1 ill. perform all indicated substitutions into the frequeni equation, I (ion A. xt |, ,1, ,d be of tc/w and make tin indies' d substi- i q Equal ion A. ■1,. , binuethepi >f selecting values of mc/w until Equation ■ 1 to desi d accuracy I t othei ilues of b d repeat the above procedure a,s nam flit d)|c. 1 procedure i shown in Fig. 2 where the velocity ,f i) ia plotted : ist frequency of vibration tinu bg de Pig 2 ■ iai iIk are thret possil w . - velocities for each : | Mm. thick) The bighesi of th< . vel it h i am ..I loi udinal v - - in ■ i pavement i<l< The low< t of tli i Ion' s i- alm< if ti versa ' ■ waves in the pa\ i mi ni i be d< . The intermediate locity is het i n * ii tin ul • .done and i i of Raj leigh in ui \i In [iiencii ill tlo< evi li< ch d: - i ' he |o ment, liglu-st and low ire p ctically indepei l- i ;ulc jh of ( l< i)ij>oi I an< < I J • l inav therefon urn rosary to ol ■ tig in (j on. 1 1 < a'l, < he follow I \ ! ) nio^hc i II he :t<lt I ng dina) _ / rvi • • iii i ■ i DYNAMIC TESTING OF PAVEMENTS 483 m Assumptions p,-- 0.8 p 6,-0.46 <D 0> where p- mass density 6 = shear modulus jx = Po is son's ratio without subscript refers to pavement with " " " subgrade Highest velocity % approximately that of longitudinal waves in a free slab 0.4 Velocity of Rayleigh waves in subgrode in term e dio te velocity f0 Lowest velocity, approximately that of_ transverse waves in a free slab Velocity of Rayleigh waves in pavement ro 0.2 0,4 0,6 Frequency times thickness times yp/G Fig. 2— Effect of frequency of vibration, thickness of pavement; and properties of subgrade on velocity of propagation of waves These two frequency equations are readily obtained from Equation A by setting G x equal to zero. When this is done, either Equation L or T must be satisfied in order that a solution exist. No correspondingly simple equation can be written for t he approxi- mate determination of the intermediate velocity. However, at low frequencies it is approximately the velocity of Rayleigh waves in the subgrade; at high frequencies it is approximately the velocity of Ray- leigh waves in the pavement; and at intermediate frequencies it lies between these two limiting velocities. Therefore, a knowledge of these two limiting velocities may be all that is necessary for practical purposes. The relations are: For Rayleigh waves in the subgrade 2 = Vd «r) (1 - &i 2 ) (Ri) and for Rayleigh waves in the pavement 2 k (R) The foregoing analysis is based upon the assumption that the proper- ties of the subgrade do not vary with depth. The properties of the subgrade, of course, do vary with depth; but, because of dampening, the properties at greater depths probably have no appreciable effect on the wave. It was shown by Love< l0 > that the velocity of Rayleigh 484 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE April 1945 waves decreased with frequency if the rigidity of the earth increased with depth. Therefore, le might expect in an actual test that the inter- mediate vel< ity would first deer* se and later incres - with increase requency rather than exhibit a continual increase a- -hown by Fig. 2. Since for a gi\ q pavement ou a given subgrade there are three pos- 8 il v ,;r lociti( - >r each frequency, there may be -ome question a situatioi in a given test. It should also be kept in mind that of these wav< locities the motion may be that corresponding - eitl he H-soluti( or th V-solutioo or a combination of tl m. Final conch - - will pro biy h to await a study of test data, but itisb tl ty P vibration can be largely controll I by the d] _ Tin- belief i^ based upon a study of the theoretical ds. This study indicates the following: (1 It th< driving is d ted horizontally, wai - of the H- . wit] ghest of the tin velocities should predominate onclus d is based upon tin t that in this case points on the 1 = have only a I rizontal re on and of relatively high ampli- i;, ; ively lai in tion of tl oint ipplication of tl driving and din n oi the driving force is neo ssary for lai s energy 2 1: thi borii illy direc d foro - not applied n r the i atral f tl . slab, v res tl lowesl the three velocities will also b 1. T » - will also be of tl H-soli Thea lower- Locity \ p; luced to the exclusion of tl a\< - of th th Sliver ] hi - a < th ] w a the driving applied 1 rizontalh p of . v - - higb 3t rather t ; n tl \ ill If t: d . .■ I vertically, wav< oi Vhs lution i v - ild predomim This ty j. 1 mo 'ii for points oi the lin< r = (4) V s of 1 ial velocity should be pi I in all y y control! pro]) th 1 that . ti\ mplitude will * la- in _ prop o: gi ]'• rhaps I ! Ill • unced h fn ienci< It is believed thai mid rodu unbalan ing i - I be ma ibl for tl of tl That h pr 1 i for both H- and V- < ith oth lieved to 1 Wav Ftl \ - lution fa - i il • tin a Id 1 pr i by a pressun bulb inseii f 1 ment if the ] ssure flucti i j ily. DYNAMIC TESTING OF PAVEMENTS 485 It should be remembered that the velocities mentioned above are the limiting velocities which are approached asymptotically as the waves get farther from the source. Since velocity varies with distance from the source, the nodes are non-uniformly spaced, especially near the origin, and spaced differently for the different types of vibration. Moreover, in vibration according to the H-solution the character of the motion and the spacing of the nodes are different in the direction of the driving force, north-south directions in Fig. 1, from what they are in a direction at right angles to the driving force, east-west in Fig. 1. The location of the nodes will also depend on which displacements, horizontal or vertical, are being detected. The table below giving 1/tt times the roots of the Bessel functions involved should be helpful because the relative distances of the nodes from the origin should correspond to the tabular values given. For example, as is evident from the equations for displacement, the vertical displacement in the V-solution is zero at every distance r from the origin for which J (ar) is zero. 1/ir Times the Roots of Bessel Functions Order of the Root 1 2 3 4 5 n Jo(x) 7655 1 757 2.755 3.754 4 753 n - 25 •/, 1 220 2 233 3.238 4 241 5 243 + 0.25 Jo - J x (x) x 5860 1.696 2.717 3 726 4.731 n - . 25 ♦Where n is a large number. As is evident from an examination of the equations for displacements, the amplitude of motion of the antinode decreases with distance from the origin. The rate of decrease depends on the angle 9 for waves of the H-solution: for example, in the direction of the driving force the amplitude is approximately inversely proportional to the square root of the distance from the origin; at right angles to the driving force the amplitude is approximately inversely proportional to the three-halves power of the distance. In the V-solution the motion of the antinodes is approximately inversely proportional to the square root of the distance away from the origin in any direction. These facts should be helpful in interpreting results. Based upon the foregoing theoretical analysis the following experi- mental procedure with equipment such as that described by Long et al. (6) is recommended. 486 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE April 1945 1. Place driver so as to produce a vertical driving force to the top of the pavement slab. 2. With input to driver set at a given frequency determine location of nodes by moving "pick-up" on radial lines from the driver and noting phase shift and magnitude of response on oscilloscope. (One pair of plates of the cathode ray oscilloscope will be connected to the pick-up circuit and the other pair will be connected to the driver circuit.) 3. Repeat the test at various frequencies. 4. Either repeat the above with driver set so as to produce horizontal motion or use the Long et al. method of determining the velocity of longitudinal waves. 5. Plot velocity of waves versus frequency and compare with curves of Fig. 2. 6. Determine proper values of G/p and thickness of slab that will give best agreement with theoretical curves. 7. If data on the properties of the subgrade are desired and the waves of intermediate velocity have not been detected and their velocity determined, use a mechanical driver of relatively low frequency. This should produce the waves of intermediate velocity and of sufficient relative amplitude for detection. Example Suppose thai at 1000 cycles per second the distances between two nodes (other than tli- irsl two nodes with the driving force acting first vertically and then horizontally are 2.25 ft. and 7.00 ft., respectively. The corresponding velocities will be 4,500 ft. pel \ and 1 1,000 fl per sec. (1 =21 times frequency/ and the ratio of the velocities w ill be 1500 1 t,000 = o 1. Assume the lower velocity to be that of transverse waves nd 1 1 j ' highei velocit} to he thai of Longitudinal waves. An examination of Fig. I ' that the velocity I these waves have a ratio of 0.H21 at a value of about 0.115 f. the ;i|.m asaG and thai the corresponding values of the ordinates are approximately i ") ah' I 0.50. Tin.- mean£ I hat , 11,000 V( p - ■ — = 9,030 it 'sec 1 .55 a i 9,030 X 0.115 thickness = = 1.04 It. 1,000 lb. I s If den* f pavement is 150 lb. | ft*, then p = 4.66 — - — od G = 4 I 9030 - = -0 X 10 6 lb. per ft-' = 2.64 X 10* psi Jl ' p - 1 <i r • n E = 1 iX 2 64 X 10» = 0.15 X 10 8 psi A I t at other fi [u< v, J000 cycles pei eco Id n apj i- matelyihe mevahi irthickne L moduli for the pavement. If radically diff< va are ol d, then th bave not 1 iden ied. One v< I in on* e ie m\\i\\\ have n the intermediate v< DYNAMIC TESTING OF PAVEMENTS 487 If the intermediate velocities are not in accord with the middle curve of Fig. 2, then other curves based on other assumptions should be prepared for these velocities. All of the above discussion has pertained to sustained vibration (stationary waves). Sustained vibration can be considered as the result of two equal continuous wave-trains (progressive waves) traveling in opposite directions. An adequate treatment of a single, impact-gener- ated wave-train of finite length traveling away from a source is beyond the scope of this paper. Because of the finite length of the train and because many different frequencies are usually represented, not all parts travel at the same velocity and the wave-form changes as the wave proceeds. However, of practical importance is the fact that the higher velocity (upper curve, Fig. 2) is almost independent of frequency for low frequencies. Therefore, a wave-train of longitudinal waves of low frequency can be propagated with relatively little change of wave-form. It is probably because of this fact that the velocities of "longitudinal" waves in pavements have been determined successfully by measuring the time required for a short wave-train of longitudinal waves to travel between two points, the short wave-train being produced by an impact. (6) The effects of variations in physical properties of the subgrade at different depths below the pavement, and the significance of the fact that the pavement does not extend indefinitely in a horizontal direction have not been considered. However, vibration of appreciable amplitude will probably not extend very far from the source either down into the subgrade or horizontally in the pavement owing to the effect of internal friction, especially at high frequencies. It is therefore believed that only the pavement within a few feet of the source and only the material immediately below the pavement will have an appreciable effect upon the wave velocity near the source when the frequency is relatively high. If the character of the subgrade at considerable depth is desired, relatively low frequencies would be required. The above analysis may be inadequate for a study of the variation in properties of the subgrade with depth since no provision was made for such variation in the equation. SUMMARY Equations are derived for the combined vibration of pavement and subgrade. Numerical solution of the frequency equation shows that for a given slab on a given subgrade three wave velocities are possible for each frequency of vibration. The highest of these velocities is almost the same as that of longitudinal vibration of a free slab; the lowest, almost the same as that of transverse (flexural) vibration of a free slab; and the intermediate, somewhere between the velocity of Rayleigh waves in the uncovered subgrade and the velocity of Rayleigh waves in the pavement. 488 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE April 1945 Suggestions are given for the production of any one of the three possible velocities to the virtual exclusion of the other two. NOTATION (PARTIAL LIST) r, Q, z = cylindrical coordinates u, v, a: = displacements in r, G, z directions, respectively, of the point (r, 0, z) in the pavement Ui, v h 10] = displacements in r, 9, z directions, respectively, of the point (r, 9, z) in the subgrade t — time p, p, = mass densities of pavement and subgrade, respectively E ~ a + n i - 2~» ' Xl ~ (1 + m ) (1 - 2 M i) By E] = Young's modulus for pa emeni and for Bubgrade, respectively M ^ ^ = poisBon J 8 ratio for pavement and subgracle, respectively G } G = Shear modulus for pavemenl and subgrade, respectively <j x a y , a z = norma] stresses Trg f TQ Zt etc. = shear sses = norma] ain in x-di] i ion , ye*, Jxy = shear e ra lb p = 2ir timi frequency of sustained vibration a = p/] = 2m I win V is the radial velocity which the wave approaches asympto- tical) ii g« te farther from its source • 1 1 is the corresponding wave Length = hall i bickn< - oi pa^ ement p I p i bio of velocitj oi : »pa bion to velocity of compressional waves n ~~ a \ x -f 2G jn unerii '' pavemenl p I p ratio of >city of propagation to velocity of si vr waves in interior ~ \ q o\ i ; ordinate of I 'ig. 1 m = a V 1 — o n = a \ 1 b 2 a lf h ■ definition! ac a, b, pn, and n, espectively, < pt that thoK with subscri] unit) r< i to subgrade instead of pavemenl R< [ i ]i u ' ' i '' ' ,n ]n brackets J . ./ J 5 of tin &rs1 kind of order zero and unity, respectively // ( \ Hi 1 = REFERENCES L T. 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